Explore the foundations of computability theory in this lecture from CMU's "Great Ideas in Theoretical Computer Science" course. Delve into Turing Machines, their formal definition, and programming exercises. Examine the concept of decidable languages and the Church-Turing Thesis. Investigate uncomputable functions and the undecidability of the Halting Problem. Learn about the historical context of computation and algorithms, including Turing's inspiration from early computers.